Quadratic twists of elliptic curves and class numbers

For positive rank r elliptic curves E(Q), we employ ideal class pairingsE(Q)×E−D(Q)→CL(−D), for quadratic twists E−D(Q) with a suitable “small y-height” rational point, to obtain explicit number lower bounds that improve on earlier work by the authors. E(a):y2=x3−a, r(a), this givesh(−D)≥110⋅|Etor(Q)|RQ(E)⋅πr(a)22r(a)Γ(r(a)2+1)⋅log⁡(D)r(a)2log⁡log⁡D, representing general improvement classical bound of Goldfeld, Gross and Zagier when r(a)≥3. We prove E−D(a)(Q) such point (resp. ≥2 under Parity Conjecture) is ≫a,εX12−ε. give infinitely many cases where r(a)≥6. These results can be viewed as an analogue estimate Gouvêa Mazur twists, in addition “log-power” improvements Goldfeld-Gross-Zagier bound.